The government has a grand plan. Since last October the Department for Education has been pushing flexible working as a solution to overcome the shortage, urging heads to offer part-time work, and serious thinkers meeting at “flexible working summits” to pledge their support.

So far, so family and life friendly. But what the flexi-worker advocates don’t seem to have realised is that more part-time working could actually increase the teacher shortage.

Teacher Tapp, an experimental app I’m involved with, surveys around 1,800 teachers each day. When asked about part-time working, 40% said they would choose to reduce their hours. At that rate, even if they reduced their week by just one day, the country would need an additional 40,000 teachers just to cover the shortfall. That’s more teachers than are trained in one year, and far more than would stay because of flexible working.

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If this sounds alarmist, look at what happened with GPs in the early 2000s. Back then, the government made working conditions more “family friendly” to encourage people into the roles. It worked. At first. Now the average GP works four days a week, with the hours they choose to work still declining, and so the shortage continues. Flexibility may be family friendly, but it doesn’t solve recruitment shortages.

The real problem of teaching is not its inflexibility. It’s the never-ending demands of the job. Not only do teachers typically spend more than 20 hours a week in front of classes (a figure that is one of the highest in the world) but there are no limits to the other work they are expected to do. It is assumed they will do whatever is required to get the outcomes demanded.

It’s no wonder that one teacher recently told me: “I’d love to work a four-day week. That way, I could spend Friday doing all the work things I normally do on a Sunday, and then spend the weekend with my kids.” The workload is so bad that teachers would be willing to work for free one day a week in return for seeing their children. You can see why the government thinks flexible working is brilliant.

But, what’s the alternative? Becky Allen, an academic at the Institute of Education, recently suggested an experiment. Each day schools would open for teachers at, say, 8am and close at 4.30pm. No work would be allowed either side. What could be done in the allotted time would show what it is possible for teachers to do, and school leaders, the government and inspectorate could recalibrate expectations accordingly.

Politically, this is difficult. No education secretary wants to preside over a reduction in service levels. But school improvement over the years has been built on the backs of teachers sacrificing their personal lives. Their numbers are diminishing, and flexible working is unlikely to boost them. In fact, it could well make the situation worse.